### Can Hedge Funds Profit from Parrondo's Paradox?

Last month, there was a flurry of blogging about hedge funds that use strategies that statistically profit in most years but eventually will blow up. Martin Wolf wrote one such column, referring to an article by Dean Foster and Peyton Young. The idea behind the article is that hedge funds are often set up (whether by design or by accident) to have good returns most years with a small chance of a blow-up each year. Therefore, the odds are that a hedge fund manager will look like a genius with great alpha, at least until the fund blows up, by which time he will have collected his or her "2 and 20" for several years. The strategy Foster and Young suggest is to long Treasury Bills and write deep out of the money put options on the S&P 500. Let's say that there is a 10% chance that the put options will be exercised in a given year. This means that it is likely that the value of the fund will grow every year by the value of the options sold plus the interest earned on the T-bills. Indeed, you could run this for five years and have a 60% chance of not blowing up in that entire period. All along, you (the hedge fund manager) are collecting your 2% of funds under management and 20% of return above a contracted benchmark.

Now obviously hedge funds aren't built so simply. The one above would seem to any reasonably sophisticated investor to be a naked, obvious scam. But Foster and Young (and Wolf) suggest that many hedge funds, though more complex, are built around similar probable returns. And the way we've been seeing so many blow up recently suggests they may be right.

But what got me thinking was their example of a fund that had a 10% chance of blowing up in any given year. This reminded me of Parrondo's Paradox. This paradox comes from game theory, and basically says that if you have two particular losing games, you can win by switching back and forth between them. A good visualization of this can be found here. Specifically, you need one game where the losses are steady and predictable, and another game in which the player wins most of the time, but loses big occasionally so that the net result is a loss. Now wait--that last one sounds just like what I was talking about above! Hmmmm.

As I understand it, the idea is to switch back and forth between the two games, and that over time, you will end up ahead.

So let's look at this from the point of view of a hedge fund investor. If such an investor invests in the fund described above, he will make a good return most years, but lose his shirt once every 10 years on average. On the other hand, if he found a hedge fund that was just an S&P index (obviously no such hedge fund exists), the investor would steadily lose 2% a year. But if the hedge fund combined the two strategies, switching off randomly between them, would it become a net winner per Parrondo's Paradox?

Well, I don't know. It may be that the transaction costs of switching would eat up any advantage you get. But there's no reason to speculate--one could, with some effort, simulate this. After all, the CBOE has sold puts on the S&P 500 for a long time (since 1989, I think) and you can get data on what deep out-of-the-money puts were available and their prices, margin requirements, and transaction costs. Likewise, we can get historical information about S&P indices and ETFs. Given this, I could write a program that simulates a fund randomly switching back and forth between the two "games" (writing puts and longing the index). The idea would be to run the simulation thousands of times and then calculate the average outcome, comparing it of course to the outcome of either just writing puts or just longing the index. I would probably want to try various start dates (or even make the start date of the simulation random), and experiment with various minimum and maximum holding times for each of the two strategies.

This would be an interesting experiment to run. The general consensus is that one can't use Parrondo's Paradox in financial situations, which is logical because it does seem to create something from nothing. But a well-wrought simulation would be worth seeing, even if it simply proved that switching randomly from longing the S&P and writing puts on it is not really a Parrondo game.

Now obviously hedge funds aren't built so simply. The one above would seem to any reasonably sophisticated investor to be a naked, obvious scam. But Foster and Young (and Wolf) suggest that many hedge funds, though more complex, are built around similar probable returns. And the way we've been seeing so many blow up recently suggests they may be right.

But what got me thinking was their example of a fund that had a 10% chance of blowing up in any given year. This reminded me of Parrondo's Paradox. This paradox comes from game theory, and basically says that if you have two particular losing games, you can win by switching back and forth between them. A good visualization of this can be found here. Specifically, you need one game where the losses are steady and predictable, and another game in which the player wins most of the time, but loses big occasionally so that the net result is a loss. Now wait--that last one sounds just like what I was talking about above! Hmmmm.

As I understand it, the idea is to switch back and forth between the two games, and that over time, you will end up ahead.

So let's look at this from the point of view of a hedge fund investor. If such an investor invests in the fund described above, he will make a good return most years, but lose his shirt once every 10 years on average. On the other hand, if he found a hedge fund that was just an S&P index (obviously no such hedge fund exists), the investor would steadily lose 2% a year. But if the hedge fund combined the two strategies, switching off randomly between them, would it become a net winner per Parrondo's Paradox?

Well, I don't know. It may be that the transaction costs of switching would eat up any advantage you get. But there's no reason to speculate--one could, with some effort, simulate this. After all, the CBOE has sold puts on the S&P 500 for a long time (since 1989, I think) and you can get data on what deep out-of-the-money puts were available and their prices, margin requirements, and transaction costs. Likewise, we can get historical information about S&P indices and ETFs. Given this, I could write a program that simulates a fund randomly switching back and forth between the two "games" (writing puts and longing the index). The idea would be to run the simulation thousands of times and then calculate the average outcome, comparing it of course to the outcome of either just writing puts or just longing the index. I would probably want to try various start dates (or even make the start date of the simulation random), and experiment with various minimum and maximum holding times for each of the two strategies.

This would be an interesting experiment to run. The general consensus is that one can't use Parrondo's Paradox in financial situations, which is logical because it does seem to create something from nothing. But a well-wrought simulation would be worth seeing, even if it simply proved that switching randomly from longing the S&P and writing puts on it is not really a Parrondo game.

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